## SUMMARY

Superficial neuromasts are structures that detect water flow on the surface
of the body of fish and amphibians. As a component of the lateral line system,
these receptors are distributed along the body, where they sense flow patterns
that mediate a wide variety of behaviors. Their ability to detect flow is
governed by their structural properties, yet the micromechanics of superficial
neuromasts are not well understood. The aim of this study was to examine these
mechanics in zebrafish (*Danio rerio*) larvae by measuring the flexural
stiffness of individual neuromasts. Each neuromast possesses a gelatinous
cupula that is anchored to hair cells by kinocilia. Using quasi-static bending
tests of the proximal region of the cupula, we found that flexural stiffness
is proportional to the number of hair cells, and consequently the number of
kinocilia, within a neuromast. From this relationship, the flexural stiffness
of an individual kinocilium was found to be 2.4×10^{–20} N
m^{2}. Using this value, we estimate that the 11 kinocilia in an
average cupula generate more than four-fifths of the total flexural stiffness
in the proximal region. The relatively minor contribution of the cupular
matrix may be attributed to its highly compliant material composition (Young's
modulus of ∼21 Pa). The distal tip of the cupula is entirely composed of
this material and is consequently predicted to be at least an order of
magnitude more flexible than the proximal region. These findings suggest that
the transduction of flow by a superficial neuromast depends on structural
dynamics that are dominated by the number and height of kinocilia.

## Introduction

The lateral line system of fish and amphibians includes two classes of mechanosensory organ on the surface of the body that function to sense water flow. The signals detected by these organs, called canal and superficial neuromasts, mediate behaviors as disparate as spawning (Satou et al., 1994), obstacle detection (Hassan, 1986) and rheotaxis (Montgomery et al., 1997). Both types of neuromast are mechanically excited by fluid forces that are transduced into receptor potentials by hair cells. The structural properties of a neuromast govern its response to flow, yet only the mechanics of canal neuromasts have been investigated in detail (Kroese and van Netten, 1989; van Netten and Kroese, 1989; van Netten and Kroese, 1987). It is consequently unclear how superficial neuromasts transduce flow or how the components of superficial neuromast morphology affect transduction. Therefore, the aim of this study was to measure the flexural stiffness (i.e. flexural rigidity or bending stiffness) of superficial neuromasts in the interest of understanding the structural basis of flow sensing in these organs.

Although canal and superficial neuromasts are similar in anatomical composition, their morphological distinctions suggest that they function differently. Both types include hair cells with kinocilia that are surrounded by a gelatinous cupula that is exposed to water flow. The pear-shaped hair cell bodies beneath the base of the cupula are surrounded by supporting cells and are innervated by efferent and afferent fibers (Dijkgraaf, 1952). However, the cupulae of superficial neuromasts protrude into the surrounding water (Fig. 1A,B), whereas canal neuromasts are recessed into a channel beneath the scales. The hemispherical cupula within this channel behaves as a rigid body that slides along the epithelium when excited by flow (van Netten and Kroese, 1989; van Netten and Kroese, 1987). In contrast, superficial neuromasts have an elongated cupula (Fig. 1B,C) that bends in flow (Schulze, 1861; Cahn and Shaw, 1962; Dinklo, 2005). Therefore, a superficial neuromast appears to operate as a cantilevered beam with a flexural stiffness that affects how mechanical information is transferred to the mechanosensory hair cells.

Subtle morphological differences in superficial neuromasts are capable of
greatly influencing their flexural stiffness. The flexural stiffness is equal
to the product (*EI*) of material and structural components
(Timoshenko, 1983). Material
stiffness, known as Young's modulus (*E*), varies by at least five
orders of magnitude among biological materials
(Wainwright et al., 1976). The
cupular material is composed of a glycosaminoglycan gel
(Sato, 1962;
Thomopolous, 1958) and may
therefore have a Young's modulus that is many orders of magnitude lower than
that of the kinocilia, which possess rigid microtubles
(Flock and Duvall, 1965). The
structural component of flexural stiffness is calculated as the second moment
of area (*I*) of a beam's cross-section, which varies with the
fourth-power of the diameter of a cylindrical beam
(Gere, 2001). This
relationship suggests that flexural stiffness may vary substantially due to
small differences in diameter. For example, a 50% increase in diameter causes
the second moment of area, and consequently flexural stiffness, to increase by
a factor of 5. Therefore, the flexural stiffness of a superficial neuromast
depends strongly on its cross-sectional size and the Young's modulus of the
kinocilia and cupular matrix.

The mechanics of superficial neuromasts are relevant to multiple areas of investigation. An understanding of these mechanics would provide a basis for functional interpretations of evolutionary (Northcutt, 1989; Webb, 1989; Webb, 1990) and ontogenetic (Appelbaum and Riehl, 1997; Blaxter and Fuiman, 1989; Poling and Fuiman, 1997; Webb and Shirey, 2003) patterns of variation in lateral line morphology among fishes. Furthermore, research on zebrafish larvae has the potential to relate neuromast dynamics to the biophysics of hair cells because this species is a major model for the study of mechanotransduction (Sidi et al., 2003; Corey et al., 2004). Additionally, an understanding of the mechanics of a superficial neuromast could assist the design of engineered flow sensors (Fan et al., 2002; Peleshanko et al., 2007).

The present study examined the contributions of the kinocilia and the cupular matrix to flexural stiffness by direct measurements. From these measurements and morphometrics, we formulated predictions of flexural stiffness along the height of the cupula. Conducting these experiments on the trunk neuromasts of zebrafish larvae allowed these structures to be visualized with Nomarski optics (Metcalfe et al., 1985) while intact on the body. The morphology of these neuromasts changes little over the course of growth (Webb and Shirey, 2003), as in other species (Munz, 1989). Therefore, our results have the potential to be applicable to the superficial neuromasts of a broad diversity of larval and adult fishes.

## Materials and methods

All experiments were conducted on zebrafish (*Danio rerio*, Hamilton
1922) larvae between 5 and 8 days post-fertilization (dpf). All larvae were
raised and maintained according to standard protocols
(Westerfield, 1995) on a 14
h/10 h light/dark cycle between 26°C and 29°C. Experiments were
performed on anesthetized larvae in a perfusion chamber with a solution of
embryo media (Brand et al.,
2002) containing 0.02% of MS-222 anesthetic (Argent, Redmond, WA,
USA). At 5 days of development, the lateral line system is composed entirely
of superficial neuromasts (Metcalfe et
al., 1985) with hair cells that produce transducer potentials
(Nicolson et al., 1998). These
cells are innervated by afferent nerves
(Alexandre and Ghysen, 1999;
Metcalfe et al., 1985), and
larvae respond to an impulsive jet of flow with a startle response
(Liu and Fetcho, 1999).
Therefore, the lateral line system appears to be fully functional by the
earliest age at which we ran our experiments.

### Morphological measurements

Cupulae were observed with a novel visualization technique (Fig. 1C,D). The trunk of an anesthetized larva was perfused with a solution of polystyrene particles (0.1μ m diameter; Polysciences Inc., Warrington, PA, USA). This particle coating made cupulae visible under polarized or Nomarski optics on a compound microscope (Zeiss Axioskope with ×40 water immersion objective with additional ×3 magnification). We verified that the innermost ring of the diffraction pattern created by these particles corresponds to the location of the cupular surface by touching the cupula with a dull probe. This technique represents an advance in methodology because past approaches for visualizing cupulae have relied on vital stains that either caused shrinkage or provided inconsistent results (Blaxter, 1984).

Digital photographs of cupulae coated with microspheres were used for
morphological measurements. Visual cross-sections of cupulae were photographed
(2080 pixels× 1542 pixels; Jenoptik, ProgRes C10 plus, Laser Optik
Systeme GmbH, Jena, Germany) and these photographs were analyzed with a
custom-made program written in Matlab (version 7.2; Mathworks, Natick, MA,
USA) that measures the position of user-selected coordinates. Cupula diameter
was measured at three positions along the height of the cupula
(Fig. 1D): at the base of the
cupula (*z*=0 μm), approximately mid-height (*z*=16 μm),
and at a position near to the distal end (*z*=32 μm). In addition,
the number of kinocilia was visually inspected and recorded at 4 μm
intervals along the height of the cupula.

### Measurement of flexural stiffness

The flexural stiffness of individual cupulae was measured in larvae restrained in an agar cast. The agar (low melting point, BP1360-100; Fisher Scientific, Waltham, MA, USA) provided a compliant surface that did not damage larvae, while firmly holding the body. This cast was created by pouring a molten solution (heated to 36°C) of 1% agar and 0.02% MS-222 in embryo media around a glass probe with a diameter that accommodates the width of the head of a larva (∼200 μm). Once cooled, the cast was submerged in embryo media (to prevent the introduction of air bubbles) and the probe was removed. Larvae were oriented on their side and pressed into the indentation created by the glass probe. This allowed the lateral surface of the body to be viewed from a fixed-stage compound microscope (Zeiss Axioskope 2FS with× 40 water-immersion objective and ×2.5 magnification cube) with transmitted illumination through the larva's body and the agar cast (Fig. 2A).

Stiffness measurements were based on recordings of the deflection of a
glass fiber as it was slowly pressed against the surface of the cupula. Glass
fibers of lengths between 6 and 9 mm were cut from cloth tape insulation
(Fisher Scientific). Both the glass fiber and the cupula were modeled as
cantilever beams with a force applied at a known distance from their clamped
bases. This is expressed mathematically by solving the Euler–Bernoulli
beam equations for the relationship between force (*F*) applied at a
distance (*d*) from its base and the deflection (δ) generated at
this position, assuming small deflections (δ/*d*<0.1)
(Gere, 2001):
(1)
In our quasi-static experiments, the force generated by the glass fiber was
equal and opposite to the elastic force of the cupula. This is expressed by
the following force-balancing relationship:
(2)
where (*EI*)_{cup}, δ_{cup} and *h* are,
respectively, the cupular flexural stiffness, cupular displacement and
position of the fiber along the height of the cupula; and
(*EI*)_{fiber}, δ_{fiber} and *l* are,
respectively, the flexural stiffness, deflection and length of the glass
fiber. The deflection of the fiber was calculated as the difference between
the displacement of its base and tip (i.e.δ
_{fiber}*=x*_{base}*–x*_{tip}),
and the deflection of the cupula was equal to the displacement of the fiber
tip (i.e. δ_{cup}*=x*_{tip}). We assumed a linear
relationship (see Eqn 2) between
the displacement of the fiber's tip and base (i.e.
*x*_{tip}=*mx*_{base}). The slope of this
relationship (*m*) was found by a least-squares linear curve fit of the
measured displacement of the tip of the glass fiber (*x*_{tip},
dependent variable) as a function of the displacement of the base
(*x*_{base}, independent variable). Given thatδ
_{fiber}/δ_{cup}=
(*x*_{base}–*x*_{tip})/*x*_{tip}=(1–*m*)/*m*,
Eqn 2 may be rewritten as:
(3)
This equation was used to calculate the flexural stiffness of a cupula from
measurements of the flexural stiffness of the glass fiber
[(*EI*)_{fiber}], its length (*l*), the position of
contact along the cupular height (*h*) and the slope (*m*).

Displacement of the base of the glass fiber (*x*_{base}) was
measured with a custom-built device. A high-precision micromanipulator (DS-4F,
Newport Corp., Mountain View, CA, USA) was used to translate the base of the
glass probe in the direction of loading. This translation was recorded with an
optical strain gauge (SPOT-2D, OSI Optoelectronics, Hawthorn, CA, USA) with a
custom-built amplifier that provided precision at the 0.01 μm level.

The position of the fiber tip (*x*_{tip}) was recorded with
an optical technique. The compound microscope (described above; see
Fig. 2B) was focused near the
distal tip of the fiber, but just proximal to its contact with the cupula. A
high-speed video camera (1024 pixels × 1024 pixels, 1000 frames
s^{–1}; 1024PCI, Photron USA Inc., San Diego, CA, USA) mounted
onto the compound microscope recorded the position of the fiber. We developed
a Matlab program to analyze these recordings in order to automatically track
the position of the edge of the glass. This program considered the pixel
intensity along a user-defined strip of pixels that spanned the edge of the
fiber in a video frame (Fig.
2A). A smoothing spline was fitted to the values of relative pixel
intensity along this strip (Fig.
2C). The program defined the edge of the fiber as the point of
most rapid positive change in intensity by finding the maximum of the first
derivative of the smoothing spline (Fig.
2D). The time history of a position recording was digitally
filtered in Matlab with a low-pass Butterworth filter with a cut-off frequency
of 1.3 Hz to remove mechanical and electrical noise. This relatively low
cut-off was necessary because of low frequency (∼2 Hz) interference by air
currents around the preparation. However, the signal was not filtered out
because loads were applied over durations of about 10 s and the transient
period during initial loading was excluded from the analysis.

Glass fibers were calibrated by calculating their flexural stiffness from
deflection measurements when loaded with a known weight. Weights small enough
to generate deflections of less than 10% were constructed from fine steel wire
(76 μm diameter) with a mass below the precision of a conventional fine
balance. This was achieved by weighing a relatively heavy length of wire (15
cm) and calculating the ratio of this weight to the length of the wire. Small
(1.3–2.8 mm) portions of the wire were then cut and their lengths were
measured from digital photographs (Canon 350D with 3456 pixels×2304
pixels on a Zeiss Stemi 2000-C microscope). The product of these lengths and
the ratio of the measured weight to length yielded weights ranging between
0.62 and 1.31 μN. The glass fibers were oriented horizontally and wires
bent into a `C' shape were placed near the tip of a fiber. The deflections of
fibers under this load were measured by digital photographs and used to
calculate their flexural stiffness (with Eqn
1). Fibers had flexural stiffness values ranging from
0.54×10^{–11} to 1.39×10^{–11} N
m^{2}.

### Error analysis

An error analysis was conducted to assess the major sources of uncertainty
in our method of measuring flexural stiffness. For each parameter that was
factored into our calculation of stiffness
(Eqn 3), we calculated random
sources of error as the standard deviation of the mean (SDOM) of repeated
measurements. SDOM assumes negligible systematic error and is equal to the
standard deviation of the measurements divided by the square root of their
sample size (Taylor, 1982).
The propagation of these errors in the calculation of flexural stiffness was
determined by examining how error in each parameter in
Eqn 3 relates to the total error,Δ
(*EI*)_{cup}, in the measurement. This yielded the
following relationship for the total proportional error in our measurements
(Taylor, 1982):
(4)
Each squared term in this equation expresses the relative contribution of a
parameter to uncertainty in our measurement of the flexural stiffness of the
cupulae. The following values for these terms were found by repeated
measurements:Δ
(*EI)*_{fiber}/(*EI)*_{fiber}=0.10,Δ
*m*/*m*(1–*m*)=0.04,
3Δ*l*/*l*<0.01, 3Δ*h*/*h*=0.06.
Therefore, error in the calibration of the stiffness of the glass fiber
contributed most to the total error in our measurements. The total error
calculated from all sources is equal to 0.12 or 12%.

A consideration of the error in our measurements influenced our
experimental methodology. The equation for total error
(Eqn 4) illustrates how absolute
errors in the slope *m* propagate in proportion to the inverse of
*m*(1–*m*). This suggests that error due to the slope can
be minimized at *m*=0.5. This slope occurs when the deflections of the
glass fiber and cupula are equivalent, which is achieved when the ratio of
flexural stiffness to the cube of length for the two structures is equivalent.
Such conditions were most easily met by adjusting the length of the glass
fiber. However, a number of experiments deviated significantly from
*m*=0.5. We therefore eliminated measurements outside of the
1/8<*m*<7/8 range in order to avoid generating excessively large
errors.

### Flexural stiffness of kinocilia

Natural variation in the number of hair cells within a neuromast provided
the opportunity to examine the effect of kinocilia on the flexural stiffness
of the cupula. All stiffness measurements were conducted on neuromasts located
in the same caudal region (P8) (Raible and
Kruse, 2000), which contained between 5 and 13 kinocilia. By
assuming that stiffness varies in proportion to the number of kinocilia, the
flexural stiffness of the proximal region of the cupula was predicted to be
the sum of the stiffness from the cupular matrix
[(*EI*)_{matrix}] and the product of the number of kinocilia
(*n*) and the flexural stiffness of an individual kinocilium
[(*EI*)_{kino}]:
(5)
We solved for (*EI*)_{kino} and (*EI*)_{matrix}
from measurements of (*EI*)_{cup} by a linear regression with
*n* as the independent variable. For this regression, we calculated the
mean values of measurements grouped by the number of hair cells. These values
were weighted by sample size in a linear least-squares curve fit
(Quinn and Keough, 2002) for
the slope [i.e. (*EI*)_{kino}] and intercept [i.e.
(*EI*)_{matrix}] of Eqn
5. Young's modulus of the matrix material was calculated by
dividing (*EI*)_{matrix} by the second moment of area for the
base of the cupula, *I*_{cup}. The second moment of area was
calculated from a measurement of the diameter of the cupula at its base
(*D*), using the following equation
(Gere, 2001):
(6)
Although stiffness measurements were conducted solely on the proximal region
of the cupula, our morphological measurements provided a means to predict how
flexural stiffness varies along the length of the cupula. The contribution of
the kinocilia to flexural stiffness at a particular height was calculated as
the product of (*EI*)_{kino} and the number of kinocilia
(*n*) at that position. The stiffness provided by the cupular matrix
along the cupular length was calculated as a product of Young's modulus of the
matrix and the second moment of area as a function of height
[Eqn 6, with *D* varying
with height, i.e. *D*(*z*)]. Values for *D*(*z*)
and *n* were calculated for heights between measurements by linear
interpolation.

## Results

The flexural stiffness of individual superficial neuromasts was measured
using the methods described above. Pressing a glass fiber against a cupula
caused both structures to deflect. These deflections were recorded by tracking
the position of the base (measured by optical strain gauge) and tip (measured
with high-speed microscopy, Fig.
2) of the glass fiber (Fig.
3A). After filtering these data, we found the slope (*m*)
describing the relationship between the displacement of the base
(*x*_{base}) and tip (*x*_{tip}) of the glass
fiber (Fig. 3B) in order to
calculate the flexural stiffness of the cupula (using
Eqn 3). This relationship was
well approximated (*r*^{2}>0.90) by linear curve fits in all
experiments. There were 15 successful experiments, all in separate individuals
(*N*=15), and these results are provided in
Table 1.

These measurements provided force–deflection curves from which an
estimate of the forces used in our experiments could be inferred. In a
representative experiment, a glass fiber pressed against a cupula with seven
kinocilia at a height of 8.4 μm generated a deflection of ∼2 μm
(Fig. 3,
Table 1). This deflection
corresponded to a force applied against the cupula of ∼0.35 nN (using
Eqn 1). These values allow us to
evaluate the implicit assumption in Eqn
1 that the cupula pivots little at its base. The base of the
cupula experiences a moment generated by the applied force (8.4μ
m×0.35 nN=2.9×10^{–15} N m), which is resisted
by the hair bundles. These bundles may be modeled as torsion springs with a
stiffness equal to that measured for canal neuromast hair cells
[2.9×10^{–14} N m rad^{–1} in the ruff,
*Acerina cernua* (van Netten and
Kroese, 1987)]. The pivot angle at the base of the cupula may be
calculated as the bending moment divided by the product of bundle stiffness
and the number of hair cells (2.9×10^{–15} N
m/(2.9×10^{–14} N m rad^{–1}×7)= 0.014
rad). This calculation yields an angle that would permit a deflection at the
tips of the kinocilia of 0.12 μm (0.014 rad×8.4 μm), which is
small (6%) relative to the measured deflection of 2 μm. Furthermore, this
calculation assumes the hair bundles to be aligned with the neutral axis of
the cupula and would predict much smaller deflections if it incorporated the
actual distribution of hair bundles over the base of the cupula.

Flexural stiffness was significantly correlated with the number of hair
cells (Fig. 4). The slope of
this relationship was interpreted as being equivalent to the flexural
stiffness of a single kinocilium (Eqn
5). We thereby determined a flexural stiffness of
(*EI*)_{kino}=2.4×10^{–21} N m^{2}
for a kinocilium with 95% confidence intervals of
*L*_{1}=1.1×10^{–21} N m^{2} and
*L*_{2}=3.7×10^{–21} N m^{2}. This
range is slightly greater than the flexural stiffness of ATP-activated
(demembranated) flagella (1.0×10^{–21} N m^{2},
Fig. 5) of sand dollar
spermatozoa (*Clypeaster japnicus*) reported in the literature
(Ishijima, 1994). The
significance of this comparison is discussed below.

The cupular matrix made a small contribution to the flexural stiffness in
the proximal region of the cupula. The matrix flexural stiffness is equal to
the intercept of the linear regression (Eqn
5) for kinocilia number and flexural stiffness
(Fig. 4). We found
(*EI*)_{matrix}*=*6.9×10^{–21} N
m^{2} with an upper 95% confidence interval of
*L*_{2}=1.8×10^{–20} N m^{2} and
lower interval of
*L*_{1}=–4.0×10^{–21} N
m^{2}. This negative value is physically impossible for a passive
structure, so the lower limit was taken as zero (*L*_{1}=0).
Using the mean value for (*EI*)_{matrix} and the mean diameter
of 8.88 μm for the cupula (see below) to calculate the second moment of
area (Eqn 6;
*I*_{cup}=3.0×10^{–22} m^{4}),
Young's modulus for the matrix material was found to be 21 Pa. An upper limit
of 53 Pa was calculated from the upper 95% confidence interval of our
measurement for (*EI*)_{matrix}.

We estimated that the flexural stiffness of a superficial neuromast
decreases along its height (Fig.
6). At the base of an average neuromast, the relatively large
diameter (8.88 μm) of the cupular matrix is predicted to generate a
flexural stiffness of 0.6×10^{–20} N m^{2}
(assuming *E*_{matrix}=21 Pa). This value is about one-fifth
the total stiffness
[(*EI*)_{cup}=3.3×10^{–20} N m^{2}]
of the proximal region (assuming 11 kinocilia and
(*EI*)_{kino}=2.4×10^{–21} N
m^{2}). Tapering in the diameter of the cupula leads to a decrease
from a mean diameter of 8.88 μm at its base to 7.2 μm at a height of 16μ
m, and 5.5 μm at a height of 32.0 μm
(Fig. 6C). This causes a
gradual reduction in stiffness generated by the matrix
(Fig. 6E). Kinocilia exhibited
a gradual decrease in number from a mean value of 11 at the base, to eight at
a height of 16 μm, while being completely absent beyond a height of 24μ
m. These reductions in the number of kinocilia strongly influence cupular
flexural stiffness (Fig. 6E).
Therefore, flexural stiffness is predicted to vary solely with the second
moment of area of the matrix in the cupular tip region. Together, the
reduction in the number of kinocilia and cupular diameter with increasing
height result in a decrease of superficial cupular flexural stiffness at the
proximal region (3.3×10^{–20} N m^{2}) to less
than an order of magnitude lower (0.2×10^{–20} N
m^{2}) at the tip region (Fig.
6E).

## Discussion

Our measurements of cupular flexural stiffness provide a basis for understanding the micromechanics of superficial neuromasts. According to our results, kinocilia dominate the flexural stiffness of the proximal region. The cupular matrix surrounds and extends beyond the kinocilia, thereby forming a highly flexible tip. This suggests that a superficial cupula functions as a two-part beam with mechanical properties that are largely determined by the height and number of kinocilia.

### The flexural stiffness of kinocilia

Kinocilia are similar to cilia and eukaryotic flagella in their ultrastructure. They possess a 9+2 arrangement of microtubles (Fawcett, 1961; Flock and Duvall, 1965) with radial spokes and outer dynein arms [Fig. 5A,B(ii)]. However, kinocilia lack nexin links and inner dynein arms [Fig. 5B(i)], which may preclude motility (Kikuchi et al., 1989). Although their ability to generate force has not been ruled out (Ross et al., 1987), kinocilia are generally regarded as passive transmitters of deflections that are transduced by channels in the stereocilia (Hudspeth and Jacobs, 1979).

Given their similarities in ultrastructure, it is informative to compare
our measurements of flexural stiffness in kinocilia with that in cilia and
flagella. Our measurements for the flexural stiffness of kinocilia
(1.1×10^{–21} to 3.7×10^{–21} N
m^{2}) are about 100 times lower than the direct measurements
(2×10^{–19} to 3×10^{–19} N
m^{2}) from gill cilia in a clam (*Mytilus edulis*) by Baba
(Baba, 1972). However, Okuno
and Hiramoto (Okuno and Hiramoto,
1979) could not replicate these results in sea urchin
(*Hemicentrotus pulcherrimus*) flagella, which showed substantially
lower flexural stiffness (∼1×10^{–20} N
m^{2}). When treated with 10 mmol l^{–1} ATP,
demembranated flagella became an order of magnitude less flexible
(∼1×10^{–21} N m^{2}), presumably because ATP
causes dynein to detach from the microtubules within the axoneme
(Okuno and Hiramoto, 1979).
This measure of stiffness was verified in sand dollar (*Clypeaster
japnicus*) spermatozoa that were loaded hydrodynamically
(Ishijima and Hiramoto, 1994).
We found this value to be slightly, but significantly, less than our
measurements for the flexural stiffness of a kinocilium
[Fig. 5B(ii)]. The greater
stiffness of kinocilia is probably a consequence of greater spacing between
their microtubule doublets [178 nm between outer edges
(Flock and Duvall, 1965)] than
in flagella (158 nm) (Brokaw,
1989).

The structural and mechanical similarities between kinocilia and flagella
raise the potential that kinocilia are polarized in their mechanics. The
present study measured flexural stiffness in the kinocilia of the P8 neuromast
(Fig. 1) by applying force in
the anterio-posterior direction. It is in this direction that the neuromast is
sensitive, due to its arrangement of stereocilia
(Lopez-Schier et al., 2004).
This direction of loading is perpendicular to the axis of the central pair of
microtubles within the kinocilium [Fig.
5B(ii)] (Flock and Duvall,
1965). Using the central pair for alignment, the kinocilia in the
present study were loaded in the same direction as the beating plane of a
flagellum [Fig. 5B(i)].
Ishijima and Hiramoto (Ishijima and
Hiramoto, 1994) found the passive stiffness of flagella along the
beating plane to be lower than when they were loaded perpendicular to this
plane [1.2×10^{–20} N m^{2},
Fig. 5B(iii)] by a factor of
12. If one assumes similar polarity, then kinocilia would be predicted to be
more than an order of magnitude more flexible in the direction in which the
hair cells sense flow (anterio-posterior, in this case) than in the
perpendicular direction (dorso-ventral).

### Young's modulus of the cupular matrix

Prior studies have suggested that the cupular matrix possesses a
biochemical composition that is grossly similar to the mesoglea of cnidarians.
Like mesoglea, the cupula is composed of a well-hydrated neutral
glycosaminoglycan gel (Sato,
1962; Thomopolous,
1958). Mesoglea commonly contains protein fibrils composed of
collagen, collagen-like and fibrillin-like molecules
(Chapman, 1953;
Grimstone et al., 1958;
Gosline, 1971;
Megill et al., 2005). The
cupula includes extracellular fibrils of unknown composition in the canal
neuromasts (Jielof et al.,
1952; Flock,
1965b; Munz, 1979;
Kelly and van Netten, 1991)
and the superficial neuromasts of some species [e.g. *Fudulus
heteroclitius* (Denny,
1937)], but not others [e.g. *Sarotherodon niloticus*
(Munz, 1979)]. Such fibrils
were not found in the superficial neuromasts of embryonic zebrafish previously
(Munz, 1979) and were not
observed in larvae with polarized or differential interference contrast
microscopy in the present study.

The relatively high compliance of the zebrafish cupula appears to be
related its lack of fibrils. Its Young's modulus (∼21 Pa) is more than two
orders of magnitude less than that of blind cavefish (∼8 kPa in
*Astyanax fasciaus*) (Peleshanko et
al., 2007), which appear to possess fibrils (in *Astyanax
hubbsi*) (Teyke, 1990). An
even greater range of Young's modulus is found in mesoglea. For example, the
high fibril density of sea anemone (*Anthopleura xanthogrammica*)
mesoglea has a Young's modulus (∼100 kPa)
(Koehl, 1977) that is four
orders of magnitude greater than that of mesoglea lacking these fibrils (e.g.
*Polycorchis penicillatus*) (Megill
et al., 2005). Mesoglea lacking fibrils from the bell of some
hydromedusae has an estimated Young's modulus (∼50 Pa)
(Megill et al., 2005) that is
similar to what we have found for the cupula of zebrafish larvae. This is not
surprising given the similar molecular composition of mesoglea and the cupular
matrix.

It is unclear whether the viscoelasticity of the cupular matrix influences its function. The viscoelastic properties of mesoglea largely determine how sea anemones respond to hydrodynamic loads over different timescales (Koehl, 1977) and dictate the resilience of the bell of swimming jellyfish (DeMont and Gosline, 1988). It therefore is conceivable that the viscous component of the cupula matrix (Peleshanko et al., 2007) could act to filter high-frequency stimuli. Alternatively, the dominance of kinocilia in the proximal region (Fig. 6) may cause the cupula to respond elastically to hydrodynamic loads. Given the quasi-static nature of the present experiments, further investigation will be necessary to resolve the role of structural viscosity in the mechanics of superficial neuromasts.

### A model for superficial neuromast mechanics

Our results suggest a model for the structural mechanics of superficial neuromasts (Fig. 7A,B). According to this model, the cupula behaves as a beam that is anchored to hair bundles, is stiff in its proximal region, and is compliant at its distal tip. The juncture at the base of the cupula behaves as a pivot that is coupled to hair bundles that act as a spring. The proximal region of the cupula has a flexural stiffness that is proportional to the number of kinocilia (Fig. 4). This stiff region extends to the height of the kinocilia, with the remainder of the cupula height providing a compliant tip. The cupula deflects when excited by flow that is governed by boundary layer hydrodynamics over the surface of the body (Jielof et al., 1952; Kuiper, 1967; Hassan, 1985; Kalmijn, 1988; Teyke, 1988; Dinklo, 2005).

This model suggests that the sensitivity of a superficial neuromast largely depends on its morphology. The hair cells within a neuromast generate transducer potentials that are proportional to the deflection of the kinocilia at low amplitudes (Flock, 1965a). Therefore, morphological properties that increase flexural stiffness serve to reduce the sensitivity of the neuromast by decreasing deflection. For example, a greater number of hair cells within a neuromast provides more kinocilia that stiffen the cupula and reduce sensitivity. Similarly, a cupula of greater diameter will have a larger second moment of area that acts to reduce deflection. However, these features may also promote sensitivity. Although a greater number of hair cells will stiffen the cupula, the neurobiological sensitivity of the neuromast will increase because there are more sensory cells. A larger cupula diameter increases flexural stiffness, but also provides a greater area for fluid forces to cause greater deflection. Therefore, trade-offs exist in the design of neuromasts that suggest the possibility that an optimal combination of morphological parameters could maximize the sensitivity of an individual neuromast. The design of neuromast arrays may alternatively benefit from a variation in morphology that creates a variety of frequency responses and sensitivities to facilitate range fractionation.

Our superficial neuromast model contrasts the micromechanics of canal neuromasts (Fig. 7C,D). Most strikingly, the structural dynamics of the cupula do not play an important role in the function of a canal neuromast. van Netten and Kroese (van Netten and Kroese, 1987) demonstrated that the cupula slides along the sensory epithelium as a rigid body. Therefore, canal neuromasts have been modeled as a rigid hemisphere that is coupled to hair bundles that function as a linear spring. This structure is driven by a uniform flow field within the canal (van Netten and Kroese, 1989) at the frequencies to which a canal neuromast is sensitive (>20 Hz). This flow varies in proportion to the difference in pressure between its pores (Denton and Gray, 1983; van Netten, 2006). In contrast, superficial neuromasts are directly exposed to a stimulus field and are sensitive to low frequency stimuli (<80 Hz) (Kroese and Schellart, 1992). At these frequencies, the boundary layer generates a spatial gradient in flow at the body's surface (Jielof et al., 1952; Kuiper, 1967; Hassan, 1985; Kalmijn, 1988; Teyke, 1988; Dinklo, 2005).

The role of structural dynamics in the frequency response of a neuromast is inextricably linked to the hydrodynamics that excite the system. The fluid forces generated by a stimulus depend on the speed and acceleration of flow relative to the motion of the cupula. This motion and, consequently, the fluid forces depend on the structural properties of the neuromast. This is illustrated mathematically by the canal neuromast model. The equations describing the hydrodynamics of the canal cupula cannot be solved without considering the stiffness of the hair bundles (van Netten and Kroese, 1989; van Netten and Kroese, 1987). Similarly, formulating a prediction for the frequency response of a superficial neuromast will require consideration of the fluid–structure interaction between the cupula and surrounding flow.

## ACKNOWLEDGEMENTS

We thank two anonymous reviewers for their helpful suggestions on the manuscript. S. Coombs made us aware of new work on the superficial neuromasts of blind cavefish. This research was supported by National Science Foundation grants to M.J.M. (IOS-0723288 and IOB-0509740).

- © The Company of Biologists Limited 2007